Epigraphically-Relaxed Linearly-Involved Generalized Moreau-Enhanced Model for Layered Mixed Norm Regularization

This paper proposes an epigraphically-relaxed linearly-involved generalized Moreau-enhanced (ER-LiGME) model for layered mixed norm regularization. Group sparse and low-rank (GSpLr)-aware modeling using ℓ₁ /nuclear-norm-based layered mixed norms has succeeded in precise high dimensional signal recovery, e.g., images and videos. Our previous work significantly expands the potential of the GSpLr-aware modeling by epigraphical relaxation (ER). It enables us to handle a (even non-proximable) deeply-layered mixed norm minimization by decoupling it into a norm and multiple epigraphical constraints (if each proximity operator is available). One problem with typical SpLr modeling is that it suffers from the underestimation effect due to the ℓ₁ and nuclear norm regularization. To circumvent this problem, LiGME penalty functions, which modify conventional sparsity and low-rankness promoting convex functions to nonconvex ones while keeping overall convexity, have been proposed conventionally. In this work, we integrate the ER technique with the LiGME model to realize deeply-layered (possibly non-proximable) mixed norm regularization and show its effectiveness in denoising and compressed sensing reconstruction.